11 thoughts on “Pythagorean Mistakes”

Two of these mistakes essentially have to do with a kid’s comfort with exponents. Exponents are important enough in high school that we need to be a lot more thoughtful about how we introduce them. The best elementary and middle school curricula have kids playing with a variety of non-algorithmic models for the four basic operations. We need to start thinking about the five basic operations instead. And we need to start developing a hierarchy of models that will come before and after a kid’s formalization of exponents as repeated multiplication.

The best elementary and middle school curricula have kids playing with a variety of non-algorithmic models for the four basic operations.

I like this. What are you thinking here? Geometric representations (squares/cubes)? In MS, we may start here but by the end of that lesson we’re multipying numbers by themselves. Thinking of “number of factors” instead? (This idea yours? Maybe Danielson’s?) Tell me more (or drop a link) about these hierarchies.

A deep association between 3, 2, and 6 getting in the way? Could be. Why 3^2 = 6 and not 5^2 = 10? This is puzzling. The inconsistency of multiplying the base by the exponent makes it easy for students/teachers to brush it off as a silly mistake (“brain fart” in my classroom) but there’s more to it than that. Even students who do not make this mistake with whole numbers will think 100^1/2 = 50 when introduced to rational exponents. We want to multiply.

Perhaps as much as the quick memorization of times tables, the quick recall (or glance at the formula chart) of the Pythagorean Theorem, without a solid conceptual understanding of what it means would lead to some easily fixable errors you present here. Great, thought-provoking questions. I like this activity.